OPTIMUM FILLING OF EMPTY BINS

In Chapter we filled empty bins by minimizing the energy output from the filtered mesh. In each case there was arbitrariness in the choice of the filter. Here we use the optimum filter, the PEF.

Geophysicists have known for many years how to compute the prediction-error filter (PEF) for a time series. Given a prediction filter, it is fairly obvious how to extrapolate the signal beyond its ends or how to fill in gaps. Classical theory focuses on instabilities that can arise, but these instabilities arise from recursive use of the prediction filter. Instability is easily avoided by use of a second stage of linear least squares to solve directly for the missing data.

Here we have code in which missing data in a 2-D data plane is restored by a two-stage linear least squares process.

In the first stage, we fit a 2-D prediction-error filter (PEF) to the given plane. Fitting equations that involve empty bins are weighted to zero before the least-squares problem is solved by conjugate direction descent. In a second stage, we take the PEF as known and find empty bin values in (or beyond) the plane so that the power out of the prediction-error filter is minimized. This two-stage method avoids the nonlinear problem we would otherwise face if we included the fitting equations containing both free data values and free filter values. Presumably, after two stages of linear least squares we are close enough to the final solution that we could switch over to the full nonlinear setup described near the end of this chapter.